График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( 6 x ) = 0 \sin{\left (6 x \right )} = 0 sin ( 6 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π Численное решение x 1 = − 3.66519142919 x_{1} = -3.66519142919 x 1 = − 3.66519142919 x 2 = − 100.007366139 x_{2} = -100.007366139 x 2 = − 100.007366139 x 3 = − 95.8185759345 x_{3} = -95.8185759345 x 3 = − 95.8185759345 x 4 = 48.171087355 x_{4} = 48.171087355 x 4 = 48.171087355 x 5 = 9.94837673637 x_{5} = 9.94837673637 x 5 = 9.94837673637 x 6 = − 51.8362787842 x_{6} = -51.8362787842 x 6 = − 51.8362787842 x 7 = 36.1283155163 x_{7} = 36.1283155163 x 7 = 36.1283155163 x 8 = − 71.733032257 x_{8} = -71.733032257 x 8 = − 71.733032257 x 9 = 92.1533845053 x_{9} = 92.1533845053 x 9 = 92.1533845053 x 10 = 80.1106126665 x_{10} = 80.1106126665 x 10 = 80.1106126665 x 11 = 34.0339204139 x_{11} = 34.0339204139 x 11 = 34.0339204139 x 12 = 26.1799387799 x_{12} = 26.1799387799 x 12 = 26.1799387799 x 13 = − 2.09439510239 x_{13} = -2.09439510239 x 13 = − 2.09439510239 x 14 = − 12.0427718388 x_{14} = -12.0427718388 x 14 = − 12.0427718388 x 15 = 14.1371669412 x_{15} = 14.1371669412 x 15 = 14.1371669412 x 16 = 68.0678408278 x_{16} = 68.0678408278 x 16 = 68.0678408278 x 17 = 31.9395253115 x_{17} = 31.9395253115 x 17 = 31.9395253115 x 18 = − 5.75958653158 x_{18} = -5.75958653158 x 18 = − 5.75958653158 x 19 = − 19.8967534727 x_{19} = -19.8967534727 x 19 = − 19.8967534727 x 20 = − 7.85398163397 x_{20} = -7.85398163397 x 20 = − 7.85398163397 x 21 = − 90.0589894029 x_{21} = -90.0589894029 x 21 = − 90.0589894029 x 22 = 18.3259571459 x_{22} = 18.3259571459 x 22 = 18.3259571459 x 23 = − 59.1666616426 x_{23} = -59.1666616426 x 23 = − 59.1666616426 x 24 = 43.9822971503 x_{24} = 43.9822971503 x 24 = 43.9822971503 x 25 = − 31.9395253115 x_{25} = -31.9395253115 x 25 = − 31.9395253115 x 26 = − 25.6563400043 x_{26} = -25.6563400043 x 26 = − 25.6563400043 x 27 = − 75.9218224618 x_{27} = -75.9218224618 x 27 = − 75.9218224618 x 28 = − 27.7507351067 x_{28} = -27.7507351067 x 28 = − 27.7507351067 x 29 = − 59.6902604182 x_{29} = -59.6902604182 x 29 = − 59.6902604182 x 30 = 62.3082542962 x_{30} = 62.3082542962 x 30 = 62.3082542962 x 31 = − 61.7846555206 x_{31} = -61.7846555206 x 31 = − 61.7846555206 x 32 = 87.9645943005 x_{32} = 87.9645943005 x 32 = 87.9645943005 x 33 = 99.4837673637 x_{33} = 99.4837673637 x 33 = 99.4837673637 x 34 = 97.9129710369 x_{34} = 97.9129710369 x 34 = 97.9129710369 x 35 = 75.9218224618 x_{35} = 75.9218224618 x 35 = 75.9218224618 x 36 = − 46.0766922527 x_{36} = -46.0766922527 x 36 = − 46.0766922527 x 37 = 12.0427718388 x_{37} = 12.0427718388 x 37 = 12.0427718388 x 38 = − 34.0339204139 x_{38} = -34.0339204139 x 38 = − 34.0339204139 x 39 = − 97.9129710369 x_{39} = -97.9129710369 x 39 = − 97.9129710369 x 40 = − 21.9911485751 x_{40} = -21.9911485751 x 40 = − 21.9911485751 x 41 = − 37.6991118431 x_{41} = -37.6991118431 x 41 = − 37.6991118431 x 42 = 78.0162175641 x_{42} = 78.0162175641 x 42 = 78.0162175641 x 43 = 21.9911485751 x_{43} = 21.9911485751 x 43 = 21.9911485751 x 44 = − 29.8451302091 x_{44} = -29.8451302091 x 44 = − 29.8451302091 x 45 = 0 x_{45} = 0 x 45 = 0 x 46 = 16.2315620435 x_{46} = 16.2315620435 x 46 = 16.2315620435 x 47 = 67.5442420522 x_{47} = 67.5442420522 x 47 = 67.5442420522 x 48 = 100.007366139 x_{48} = 100.007366139 x 48 = 100.007366139 x 49 = − 17.8023583703 x_{49} = -17.8023583703 x 49 = − 17.8023583703 x 50 = − 384.845100065 x_{50} = -384.845100065 x 50 = − 384.845100065 x 51 = − 68.0678408278 x_{51} = -68.0678408278 x 51 = − 68.0678408278 x 52 = 84.2994028713 x_{52} = 84.2994028713 x 52 = 84.2994028713 x 53 = 50.2654824574 x_{53} = 50.2654824574 x 53 = 50.2654824574 x 54 = − 49.7418836818 x_{54} = -49.7418836818 x 54 = − 49.7418836818 x 55 = − 63.879050623 x_{55} = -63.879050623 x 55 = − 63.879050623 x 56 = − 93.7241808321 x_{56} = -93.7241808321 x 56 = − 93.7241808321 x 57 = 28.2743338823 x_{57} = 28.2743338823 x 57 = 28.2743338823 x 58 = − 43.9822971503 x_{58} = -43.9822971503 x 58 = − 43.9822971503 x 59 = − 41.8879020479 x_{59} = -41.8879020479 x 59 = − 41.8879020479 x 60 = − 81.6814089933 x_{60} = -81.6814089933 x 60 = − 81.6814089933 x 61 = − 90.5825881785 x_{61} = -90.5825881785 x 61 = − 90.5825881785 x 62 = 72.2566310326 x_{62} = 72.2566310326 x 62 = 72.2566310326 x 63 = 46.0766922527 x_{63} = 46.0766922527 x 63 = 46.0766922527 x 64 = − 65.9734457254 x_{64} = -65.9734457254 x 64 = − 65.9734457254 x 65 = 90.0589894029 x_{65} = 90.0589894029 x 65 = 90.0589894029 x 66 = − 73.8274273594 x_{66} = -73.8274273594 x 66 = − 73.8274273594 x 67 = − 56.025068989 x_{67} = -56.025068989 x 67 = − 56.025068989 x 68 = 56.025068989 x_{68} = 56.025068989 x 68 = 56.025068989 x 69 = − 82.2050077689 x_{69} = -82.2050077689 x 69 = − 82.2050077689 x 70 = 96.3421747101 x_{70} = 96.3421747101 x 70 = 96.3421747101 x 71 = − 24.0855436775 x_{71} = -24.0855436775 x 71 = − 24.0855436775 x 72 = 94.2477796077 x_{72} = 94.2477796077 x 72 = 94.2477796077 x 73 = − 9.94837673637 x_{73} = -9.94837673637 x 73 = − 9.94837673637 x 74 = 82.2050077689 x_{74} = 82.2050077689 x 74 = 82.2050077689 x 75 = − 15.7079632679 x_{75} = -15.7079632679 x 75 = − 15.7079632679 x 76 = 93.2005820565 x_{76} = 93.2005820565 x 76 = 93.2005820565 x 77 = − 78.0162175641 x_{77} = -78.0162175641 x 77 = − 78.0162175641 x 78 = 65.9734457254 x_{78} = 65.9734457254 x 78 = 65.9734457254 x 79 = 10.471975512 x_{79} = 10.471975512 x 79 = 10.471975512 x 80 = − 83.7758040957 x_{80} = -83.7758040957 x 80 = − 83.7758040957 x 81 = 38.2227106187 x_{81} = 38.2227106187 x 81 = 38.2227106187 x 82 = 58.1194640914 x_{82} = 58.1194640914 x 82 = 58.1194640914 x 83 = 40.3171057211 x_{83} = 40.3171057211 x 83 = 40.3171057211 x 84 = − 39.7935069455 x_{84} = -39.7935069455 x 84 = − 39.7935069455 x 85 = 2.09439510239 x_{85} = 2.09439510239 x 85 = 2.09439510239 x 86 = 70.1622359302 x_{86} = 70.1622359302 x 86 = 70.1622359302 x 87 = − 85.8701991981 x_{87} = -85.8701991981 x 87 = − 85.8701991981 x 88 = 4.18879020479 x_{88} = 4.18879020479 x 88 = 4.18879020479 x 89 = − 53.9306738866 x_{89} = -53.9306738866 x 89 = − 53.9306738866 x 90 = 6.28318530718 x_{90} = 6.28318530718 x 90 = 6.28318530718 x 91 = − 87.9645943005 x_{91} = -87.9645943005 x 91 = − 87.9645943005 x 92 = 24.0855436775 x_{92} = 24.0855436775 x 92 = 24.0855436775 x 93 = 53.9306738866 x_{93} = 53.9306738866 x 93 = 53.9306738866 x 94 = 34.5575191895 x_{94} = 34.5575191895 x 94 = 34.5575191895 x 95 = 60.2138591938 x_{95} = 60.2138591938 x 95 = 60.2138591938
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ⋅ 6 ) \sin{\left (0 \cdot 6 \right )} sin ( 0 ⋅ 6 ) f ( 0 ) = 0 f{\left (0 \right )} = 0 f ( 0 ) = 0 (0, 0)
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left (x \right )} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left (x \right )} = d x d f ( x ) = Первая производная 6 cos ( 6 x ) = 0 6 \cos{\left (6 x \right )} = 0 6 cos ( 6 x ) = 0 Решаем это уравнение x 1 = π 12 x_{1} = \frac{\pi}{12} x 1 = 12 π x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π pi
(--, 1)
12 pi
(--, -1)
4 Интервалы возрастания и убывания функции: x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π x 2 = π 12 x_{2} = \frac{\pi}{12} x 2 = 12 π (-oo, pi/12] U [pi/4, oo) [pi/12, pi/4]
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left (x \right )} = d x 2 d 2 f ( x ) = Вторая производная − 36 sin ( 6 x ) = 0 - 36 \sin{\left (6 x \right )} = 0 − 36 sin ( 6 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π Интервалы выпуклости и вогнутости: (-oo, 0] U [pi/6, oo) [0, pi/6]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin ( 6 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \sin{\left (6 x \right )} = \langle -1, 1\rangle x → − ∞ lim sin ( 6 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ sin ( 6 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \sin{\left (6 x \right )} = \langle -1, 1\rangle x → ∞ lim sin ( 6 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(6*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x sin ( 6 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \sin{\left (6 x \right )}\right) = 0 x → − ∞ lim ( x 1 sin ( 6 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x sin ( 6 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \sin{\left (6 x \right )}\right) = 0 x → ∞ lim ( x 1 sin ( 6 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( 6 x ) = − sin ( 6 x ) \sin{\left (6 x \right )} = - \sin{\left (6 x \right )} sin ( 6 x ) = − sin ( 6 x ) sin ( 6 x ) = − − 1 sin ( 6 x ) \sin{\left (6 x \right )} = - -1 \sin{\left (6 x \right )} sin ( 6 x ) = − − 1 sin ( 6 x ) 